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Magnetism 1) In class we obtained the formula (m) m mart tanh( (1) kT for a spin 1/2 particle with magnetic moment m in a magnetic field H. a) Develop a similar formula for a spin 1 particle (Sz = 1,0,-1) and a spin 3/2 particle (Sz = 3/2, 1/2, -1/2, -3/2). Remember the magnetic moment is m gugs b) In fact a general formula has been worked out for any S. The average magnetic moment for a spin S particle is found in terms of the Brillouin function, Bs(x) according to (m) m mert The Brillouin function is given by 1 (2) 2S 2S 2J c) It is rather painful to work out this formula in general. Show that the general formula in Eq (2) reduces to Eq (1) in the case of S = 1/2. Hint: This last thing takes some algebra! Expand out the ctnh in terms of e+x and c and combine the two terms in B(x). And don't give up. 2) Suppose we are not dealing with a quantum system but with a classical magnetic moment m. (This is the case if you have a small magnetic particle that has a net magnetic moment, but the orientation can vary) In this case the statistical mechanics required to find in a field H (also along Z) is not about a sum over states, but an integral over all possible angles. a) Consider an external field H along the Z direction and use m, mcos0 The sum over states to find involves an integral over all angles as stated above. Carry out this calculation and show that < m. mL(x) Where L(x) coth(x) 1/x And x mH/kT as usual. Now, suppose you have a completely magnetized iron sphere with a radius of 5 nm. (Remember Fe has a Tc slightly above 1,000 K so at room temperature and below it is pretty much fully magnetized.) b) Estimate the total magnetic moment of the sphere remember Fe has a moment of 2.2 18 per atom c) If you put the sphere in a 1,000 Oe field, what is T m as a function of temperature. Make a plot of your results from T = 0 to T = 400K. This is what you would actually measure in an experiment. d) Is the result in (c) similar to the result you would find if you measured a much bigger sphere? How and why would it change?

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